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This essential trains for: AMC-10, AMC-12, AIME.

A perpendicular bisector of a triangle is a line that is the perpendicular bisector of a side of the triangle.

Attention: in a scalene triangle, the perpendicular bisector of a side does not generally pass through the opposite vertex!

Since any two non-parallel lines in a plane will intersect, two of the perpendicular bisectors of a triangle will have a common point. Let us denote it with K:

What information do we have about K?

Since K is on the perpendicular bisector of the side AB, it is equidistant from A and from B:

Since K is on the perpendicular bisector of the side BC, it is equidistant from B and from C:

By transitivity,

This means that K is equidistant from the points A and from C. Therefore, it is a point on the perpendicular bisector of the side AC.

The perpendicular bisectors of the sides of a triangle are concurrent (i.e., they have a common point).

The point of intersection of the perpendicular bisectors is equidistant from all the vertices of the triangle. Therefore, it is the center of a circle that passes through all the vertices - the circumcenter.

Since this point is uniquely determined, then there is a unique circle that passes through three non-collinear points.

Reciprocally, given three non-collinear points, we can find the circle that passes through them by constructing any two segments formed by these points and their perpendicular bisectors. The intersection is the center of the circle. The distance from the intersection to any of the points is the radius.

Fact: The circumcenter of a right angle triangle is the midpoint of the hypotenuse.

This fact follows from Thales theorem: if a right angle triangle is inscribed in a circle, the hypotenuse is a diameter. Therefore, the midpoint of the hypotenuse is the center.